Towards nonsymmetric conic optimization

In this paper we propose a new interior-point method, which is based on an extension of the ideas of self-scaled optimization to the general cones. We suggest using the primal correction process to find a {\em scaling point}. This point is used to compute a strictly feasible primal-dual pair by simple projection. Then, we define … Read more

On the optimal parameter of a self-concordant barrier over a symmetric cone

The properties of the barrier F(x)=-log(det(x)), defined over the cone of squares of an Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, it is proved … Read more

On Conically Ordered Convex Programs

In this paper we study a special class of convex optimization problems called {\em conically ordered convex programs}\/ (COCP), where the feasible region is given as the level set of a vector-valued nonlinear mapping, expressed as a nonnegative combination of convex functions. The nonnegativity of the vectors is defined using a pre-described conic ordering. The … Read more

”Cone-Free” Primal-Dual Path-Following and Potential Reduction Polynomial Time Interior-Point Methods

We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the … Read more

Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers

We study homogeneous convex cones. We first characterize the extreme rays of such cones in the context of their primal construction (due to Vinberg) and also in the context of their dual construction (due to Rothaus). Then, using these results, we prove that every homogeneous cone is facially exposed. We provide an alternative proof of … Read more

Improving complexity of structured convex optimization problems using self-concordant barriers

The purpose of this paper is to provide improved complexity results for several classes of structured convex optimization problems using to the theory of self-concordant functions developed in [2]. We describe the classical short-step interior-point method and optimize its parameters in order to provide the best possible iteration bound. We also discuss the necessity of … Read more